Question

December 8, 2018 WORK ALL PROBLEMS. SHOW WORK & INDICATE REASONING \ 1.) Let σ-(13524)(2376)(4162)(3745). Express σ as a product of disjoint cycles Express σ as a product of 2 cycles. Determine the inverse of σ. Determine the order of ơ. Determine the orbits of ơ 2) Let ф : G H be a homomorphism from group G to group H. Show that G is. one-to-one if and only if Ker(p) (ed) 3.) Determine, or explain why no such homomorphism exists. if possible, a nontrivial homomorphism between the indicated groups Recall that, As is the 4.) Give a clear and precise definition of each of the following terms. group of even permutations on a set of four elements ) integral domain b) ring homomorphism e) ideal d) group automorphism e) coset 5.) Determine all normal subgroups of the group Ds. Clearly indicate why each of these 2 t subgroups is normal and why these are the only normal subgbroups of Da 6.) Show that every finite integral domain is a field. o 7.) Determine all 0 divisors, if any, in each of the following rings. ne b) M(2,Z2)) Recall that M (2,22) is the ring of 2 × 2 matrices with entries in Z2 and matrix addition and multiplication mod 2. 8.) Describe up to isomorphism all Abelian groups of order 360. Indicate how you know tha these are all such groups and that the groups which you list are isomorphi cally distinct. BONUS PROBLEM (Worth possible 10 additional points.) Let a and b be elements of the group G. Show that the order of ab is the same as the order of ba. Note that it is not assumed that G is Abelian or that any of a, b, ab or ba has finite order.

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Answer 1

11) (1 3 524) (2了76)(4162)(3 745) 42(3657 Explaination w = (了 구 "i=(4-62

a, aa, a : a, a, a a4 วุโ9w.invene 옇